metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.145D6, C6.742+ 1+4, C6.932- 1+4, Dic3⋊D4⋊43C2, D6⋊3Q8⋊34C2, (C2×D4).113D6, C4.4D4⋊17S3, (C2×Q8).108D6, C22⋊C4.38D6, C42⋊7S3⋊31C2, C23.9D6⋊47C2, C2.54(D4○D12), (C2×C6).228C24, D6⋊C4.73C22, C2.54(Q8○D12), C12.6Q8⋊29C2, C23.14D6⋊35C2, C2.78(D4⋊6D6), (C2×C12).633C23, (C4×C12).222C22, (C6×D4).213C22, (C2×D12).35C22, C4⋊Dic3.52C22, (C22×C6).58C23, C23.60(C22×S3), (C6×Q8).131C22, Dic3.D4⋊43C2, C23.21D6⋊28C2, Dic3⋊C4.84C22, C22.249(S3×C23), (C2×Dic6).39C22, (C22×S3).100C23, C3⋊4(C22.56C24), (C2×Dic3).118C23, C6.D4.60C22, (C22×Dic3).147C22, (C3×C4.4D4)⋊20C2, (S3×C2×C4).123C22, (C2×C4).201(C22×S3), (C2×C3⋊D4).66C22, (C3×C22⋊C4).69C22, SmallGroup(192,1243)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.145D6
G = < a,b,c,d | a4=b4=1, c6=d2=b2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=b-1, dbd-1=a2b-1, dcd-1=c5 >
Subgroups: 608 in 220 conjugacy classes, 91 normal (31 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, C6×D4, C6×Q8, C22.56C24, C12.6Q8, C42⋊7S3, Dic3.D4, C23.9D6, Dic3⋊D4, C23.21D6, C23.14D6, D6⋊3Q8, C3×C4.4D4, C42.145D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, 2- 1+4, S3×C23, C22.56C24, D4⋊6D6, D4○D12, Q8○D12, C42.145D6
►On 96 points
(1 26 39 79)(2 74 40 33)(3 28 41 81)(4 76 42 35)(5 30 43 83)(6 78 44 25)(7 32 45 73)(8 80 46 27)(9 34 47 75)(10 82 48 29)(11 36 37 77)(12 84 38 31)(13 54 86 72)(14 67 87 49)(15 56 88 62)(16 69 89 51)(17 58 90 64)(18 71 91 53)(19 60 92 66)(20 61 93 55)(21 50 94 68)(22 63 95 57)(23 52 96 70)(24 65 85 59)
(1 67 7 61)(2 62 8 68)(3 69 9 63)(4 64 10 70)(5 71 11 65)(6 66 12 72)(13 78 19 84)(14 73 20 79)(15 80 21 74)(16 75 22 81)(17 82 23 76)(18 77 24 83)(25 92 31 86)(26 87 32 93)(27 94 33 88)(28 89 34 95)(29 96 35 90)(30 91 36 85)(37 59 43 53)(38 54 44 60)(39 49 45 55)(40 56 46 50)(41 51 47 57)(42 58 48 52)
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 6 7 12)(2 11 8 5)(3 4 9 10)(13 93 19 87)(14 86 20 92)(15 91 21 85)(16 96 22 90)(17 89 23 95)(18 94 24 88)(25 32 31 26)(27 30 33 36)(28 35 34 29)(37 46 43 40)(38 39 44 45)(41 42 47 48)(49 72 55 66)(50 65 56 71)(51 70 57 64)(52 63 58 69)(53 68 59 62)(54 61 60 67)(73 84 79 78)(74 77 80 83)(75 82 81 76)
G:=sub<Sym(96)| (1,26,39,79)(2,74,40,33)(3,28,41,81)(4,76,42,35)(5,30,43,83)(6,78,44,25)(7,32,45,73)(8,80,46,27)(9,34,47,75)(10,82,48,29)(11,36,37,77)(12,84,38,31)(13,54,86,72)(14,67,87,49)(15,56,88,62)(16,69,89,51)(17,58,90,64)(18,71,91,53)(19,60,92,66)(20,61,93,55)(21,50,94,68)(22,63,95,57)(23,52,96,70)(24,65,85,59), (1,67,7,61)(2,62,8,68)(3,69,9,63)(4,64,10,70)(5,71,11,65)(6,66,12,72)(13,78,19,84)(14,73,20,79)(15,80,21,74)(16,75,22,81)(17,82,23,76)(18,77,24,83)(25,92,31,86)(26,87,32,93)(27,94,33,88)(28,89,34,95)(29,96,35,90)(30,91,36,85)(37,59,43,53)(38,54,44,60)(39,49,45,55)(40,56,46,50)(41,51,47,57)(42,58,48,52), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,93,19,87)(14,86,20,92)(15,91,21,85)(16,96,22,90)(17,89,23,95)(18,94,24,88)(25,32,31,26)(27,30,33,36)(28,35,34,29)(37,46,43,40)(38,39,44,45)(41,42,47,48)(49,72,55,66)(50,65,56,71)(51,70,57,64)(52,63,58,69)(53,68,59,62)(54,61,60,67)(73,84,79,78)(74,77,80,83)(75,82,81,76)>;
G:=Group( (1,26,39,79)(2,74,40,33)(3,28,41,81)(4,76,42,35)(5,30,43,83)(6,78,44,25)(7,32,45,73)(8,80,46,27)(9,34,47,75)(10,82,48,29)(11,36,37,77)(12,84,38,31)(13,54,86,72)(14,67,87,49)(15,56,88,62)(16,69,89,51)(17,58,90,64)(18,71,91,53)(19,60,92,66)(20,61,93,55)(21,50,94,68)(22,63,95,57)(23,52,96,70)(24,65,85,59), (1,67,7,61)(2,62,8,68)(3,69,9,63)(4,64,10,70)(5,71,11,65)(6,66,12,72)(13,78,19,84)(14,73,20,79)(15,80,21,74)(16,75,22,81)(17,82,23,76)(18,77,24,83)(25,92,31,86)(26,87,32,93)(27,94,33,88)(28,89,34,95)(29,96,35,90)(30,91,36,85)(37,59,43,53)(38,54,44,60)(39,49,45,55)(40,56,46,50)(41,51,47,57)(42,58,48,52), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,6,7,12)(2,11,8,5)(3,4,9,10)(13,93,19,87)(14,86,20,92)(15,91,21,85)(16,96,22,90)(17,89,23,95)(18,94,24,88)(25,32,31,26)(27,30,33,36)(28,35,34,29)(37,46,43,40)(38,39,44,45)(41,42,47,48)(49,72,55,66)(50,65,56,71)(51,70,57,64)(52,63,58,69)(53,68,59,62)(54,61,60,67)(73,84,79,78)(74,77,80,83)(75,82,81,76) );
G=PermutationGroup([[(1,26,39,79),(2,74,40,33),(3,28,41,81),(4,76,42,35),(5,30,43,83),(6,78,44,25),(7,32,45,73),(8,80,46,27),(9,34,47,75),(10,82,48,29),(11,36,37,77),(12,84,38,31),(13,54,86,72),(14,67,87,49),(15,56,88,62),(16,69,89,51),(17,58,90,64),(18,71,91,53),(19,60,92,66),(20,61,93,55),(21,50,94,68),(22,63,95,57),(23,52,96,70),(24,65,85,59)], [(1,67,7,61),(2,62,8,68),(3,69,9,63),(4,64,10,70),(5,71,11,65),(6,66,12,72),(13,78,19,84),(14,73,20,79),(15,80,21,74),(16,75,22,81),(17,82,23,76),(18,77,24,83),(25,92,31,86),(26,87,32,93),(27,94,33,88),(28,89,34,95),(29,96,35,90),(30,91,36,85),(37,59,43,53),(38,54,44,60),(39,49,45,55),(40,56,46,50),(41,51,47,57),(42,58,48,52)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,6,7,12),(2,11,8,5),(3,4,9,10),(13,93,19,87),(14,86,20,92),(15,91,21,85),(16,96,22,90),(17,89,23,95),(18,94,24,88),(25,32,31,26),(27,30,33,36),(28,35,34,29),(37,46,43,40),(38,39,44,45),(41,42,47,48),(49,72,55,66),(50,65,56,71),(51,70,57,64),(52,63,58,69),(53,68,59,62),(54,61,60,67),(73,84,79,78),(74,77,80,83),(75,82,81,76)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4E | 4F | ··· | 4K | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 12 | 12 | 2 | 4 | ··· | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | 2+ 1+4 | 2- 1+4 | D4⋊6D6 | D4○D12 | Q8○D12 |
| kernel | C42.145D6 | C12.6Q8 | C42⋊7S3 | Dic3.D4 | C23.9D6 | Dic3⋊D4 | C23.21D6 | C23.14D6 | D6⋊3Q8 | C3×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C6 | C6 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 1 | 2 | 2 | 2 |
Matrix representation of C42.145D6 ►in GL8(𝔽13)
| 6 | 6 | 9 | 9 | 0 | 0 | 0 | 0 |
| 3 | 3 | 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 7 | 10 | 7 | 0 | 0 | 0 | 0 |
| 10 | 0 | 10 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 12 | 11 | 12 | 0 | 0 | 0 | 0 |
| 5 | 5 | 1 | 0 | 0 | 0 | 0 | 0 |
| 4 | 5 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 11 |
| 8 | 8 | 3 | 8 | 0 | 0 | 0 | 0 |
| 8 | 8 | 8 | 3 | 0 | 0 | 0 | 0 |
| 5 | 0 | 5 | 5 | 0 | 0 | 0 | 0 |
| 5 | 5 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 8 | 8 | 8 | 3 | 0 | 0 | 0 | 0 |
| 8 | 8 | 3 | 8 | 0 | 0 | 0 | 0 |
| 5 | 0 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 5 | 5 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(8,GF(13))| [6,3,0,10,0,0,0,0,6,3,7,0,0,0,0,0,9,0,10,10,0,0,0,0,9,9,7,7,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0],[0,12,5,4,0,0,0,0,0,12,5,5,0,0,0,0,12,11,1,1,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,2,9,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,0,2,9,0,0,0,0,0,0,4,11],[8,8,5,5,0,0,0,0,8,8,0,5,0,0,0,0,3,8,5,5,0,0,0,0,8,3,5,5,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0],[8,8,5,0,0,0,0,0,8,8,0,5,0,0,0,0,8,3,5,5,0,0,0,0,3,8,5,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,12] >;
C42.145D6 in GAP, Magma, Sage, TeX
C_4^2._{145}D_6 % in TeX
G:=Group("C4^2.145D6"); // GroupNames label
G:=SmallGroup(192,1243);
// by ID
G=gap.SmallGroup(192,1243);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,1571,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^5>;
// generators/relations